Geodesic
The cohomology ring of the real locus of the moduli space of stable curves of genus $0$ with marked points
Pavel Etingof1 ; André Henriques2 ; Joel Kamnitzer3 ; Eric M. Rains4
1 Massachusetts Institute of Technology<br/>Department of Mathematics<br/>77 Massachusetts Avenue<br/>Cambridge, MA 02139-4307<br/>United States
2 Universiteit Utrecht<br/>Mathematisch Instituut<br/>Postbus 80 010<br/>3508 TA Utrecht<br/>The Netherlands
3 Department of Mathematics<br/>University of Toronto<br/>Room 6290, 40 St. George Street<br/>Toronto, Ontario M5S 2E4<br/>Canada
4 Mathematics 253-37<br/>California Institute of Technology<br/>Pasadena, CA 91125<br/>United States
Annals of mathematics, Tome 171 (2010) no. 2, pp. 731-777.

Voir la notice de l'article provenant de la source Annals of Mathematics website

We compute the Poincaré polynomial and the cohomology algebra with rational coefficients of the manifold $M_n$ of real points of the moduli space of algebraic curves of genus $0$ with $n$ labeled points. This cohomology is a quadratic algebra, and we conjecture that it is Koszul. We also compute the $2$-local torsion in the cohomology of $M_n$. As was shown by the fourth author, the cohomology of $M_n$ does not have odd torsion, so that the above determines the additive structure of the integral homology and cohomology. Further, we prove that the rational homology operad of $M_n$ is the operad of 2-Gerstenhaber algebras, which is closely related to the Hanlon-Wachs operad of $2$-Lie algebras (generated by a ternary bracket). Finally, using Drinfeld’s theory of quantization of coboundary Lie quasibialgebras, we show that a large series of representations of the quadratic dual Lie algebra $L_n$ of $H^*(M_n,Q)$ (associated to such quasibialgebras) factors through the the natural projection of $L_n$ to the associated graded Lie algebra of the prounipotent completion of the fundamental group of $M_n$. This leads us to conjecture that the said projection is an isomorphism, which would imply a formula for lower central series ranks of the fundamental group. On the other hand, we show that the spaces $M_n$ are not formal starting from $n=6$.
DOI : 10.4007/annals.2010.171.731

Pavel Etingof 1 ; André Henriques 2 ; Joel Kamnitzer 3 ; Eric M. Rains 4

1 Massachusetts Institute of Technology<br/>Department of Mathematics<br/>77 Massachusetts Avenue<br/>Cambridge, MA 02139-4307<br/>United States
2 Universiteit Utrecht<br/>Mathematisch Instituut<br/>Postbus 80 010<br/>3508 TA Utrecht<br/>The Netherlands
3 Department of Mathematics<br/>University of Toronto<br/>Room 6290, 40 St. George Street<br/>Toronto, Ontario M5S 2E4<br/>Canada
4 Mathematics 253-37<br/>California Institute of Technology<br/>Pasadena, CA 91125<br/>United States 
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     author = {Pavel Etingof and Andr\'e Henriques and Joel Kamnitzer and Eric M. Rains},
     title = {The cohomology ring of the real locus  of the moduli space of stable curves  of genus $0$ with marked points},
     journal = {Annals of mathematics},
     pages = {731--777},
     publisher = {mathdoc},
     volume = {171},
     number = {2},
     year = {2010},
     doi = {10.4007/annals.2010.171.731},
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     url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.731/}
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Pavel Etingof; André Henriques; Joel Kamnitzer; Eric M. Rains. The cohomology ring of the real locus  of the moduli space of stable curves  of genus $0$ with marked points. Annals of mathematics, Tome 171 (2010) no. 2, pp. 731-777. doi : 10.4007/annals.2010.171.731. http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.731/

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